Atividade 1_J Chem Educ 1966_43_283

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Resource Papers-V 
Prep.& under the aponrorrhip of 
The Advisory Council on College Chemistry 
Today the teaching of chemistry prob- 
ably leans more heavily on the theory of atomic struc- 
ture than on any other single pillar. We use the con- 
cepts of shell structure to develop the periodic table 
and all the characteristic physical and chemical proper- 
ties associated with chemical periodicity. The Bohr 
model of atomic shells gives us a universal tool to 
estimate and exhibit the magnitudes characteristic of 
virtually all atomic and molecular properties. Orbitals 
are a t the base of our interpretations of molecular 
structure, and we almost always express these orbitals 
in terms of, or a t least in relation to, the orbitals of the 
constituent atoms. We even can begin to use orbital 
concepts to interpret many reaction mechanisms. 
The subject is, in fact, so integrated into our whole 
approach to chemistry that we are astonished when a 
freshman comes to us from a high school chemistry 
course that did not interpret chemistry in terms of 
orbitals. 
R. Stephen Berry1 
The University of Chicago 
Chicago, Illinois 
I. Atomic Orders of Magnitude and the Bohr Alom 
Atomic Orbitals 
The Bohr-Sommerfeld theory of atomic planetary 
orbits was the first quantitative statement of atomic 
shell structure, and is still the source of much of our 
intuition about atoms. Even more important, this 
model was the statement of the postulate of stationary 
states, a statement that simply defied the laws of classi- 
cal physics: an electron in a "Bohr atom" remains in 
orbit forever, and does not spiral in toward the nucleus. 
Classically, the angular acceleration of such a bound 
electron would force it to radiate its energy away slowly 
and eventually fall into its attracting nucleus. With 
the postulate of stationary states and the postulate of 
the quantization of action, the theory of circular Bohr 
orbits can be developed in afew lines of algebra see refer- 
ences (21 -34; also see Appendix). From it, we develop 
a powerful little table (see Table 1) of numerical expres- 
sions for the radius, velocity, energy, and period of the 
circular orbit characterized by two numbers: 2, the 
number (or effective number) of positive unit charges 
attracting the electron of interest, and n, the quantum 
number (or effective quantum number) characterizing 
that particular stable orbit. (We shall say more about 
Table 1. Expressions for Characteristic Properties of 
Circular Bohr Orbits 
Quantity Expression Value 
Zs 109,687 cm-I 
Energy, E 2h' n' or 13.58 ev or 2.178 X 10-LLerg 
tLZ nz n2 
Radius, r - e2m X 0.529 X X z em 
ez Z Z 
Velocity, V t- X 2 188 X lo8 X ; cm/sec 
2rha n8 n3 
Period X Za l * S X X Z, see 
tL = Planck's constant h divided by 2r , e = charge on the e lec 
tron, m = mass of the electron, and Ze = nuclear charge. 
effective charges and effective quantum numbers fur- 
ther along.) The Bohr-Sommerfeld model is admit- 
tedly inconsistent with classical mechanics and it 
gives some results that do not agree well with experi- 
ment, or that simulv are w r o n ~ . ~ Nevertheless its 
Alfred P. Sloan Fellow. 
' One example is the nonzero angular momentum of the ground 
state of hydrogen, and the properties such as magnetic moments 
associated with angular momentum. The theory also prohibits 
the electron from entering the nucleus; electrons actually can 
penetrate nuclei. The ability of electrons to approach and 
penetrate nuclei to varying degrees is the reason that proton mag- 
netic resonance lines occur a t s. varietu of energies in a given 
field. Without this property, nuclear magnetic resonance would 
not he anything like the powerful analytical tool that i t actually 
is. 
Volume 43, Number 6, June 7 966 / 283 
utility as a tool for estimating orders of magnitude is 
universally recognized, and it is surely the source of 
much of our visceral intuition about atomic ~t ructure .~ 
With it, we can say, for example, that phenomena oc- 
curring in times longer than about 10-l5 sec can best be 
described in terms of the time-averaged distribution 
of an electron and not by a moving point-charge, 
simply because the classical electron would go through 
many orbits during the event. Phenomena requiring 
less than lo-" see, say, are not well described by 
average distributions of charge in atoms. By the same 
token phenomena involving energies much greater than 
1&20 ev are necessarily quite disruptive to the outer 
shells of atoms, but energies a hundredfold smaller do 
not affect these shells very much. 
Naturally, the classical estimates we have just made 
have their parallels in quantum mechanics, in terms of 
wave structure and the uncertainty principle. We shall 
examine these; hut first let us spend a moment examin- 
ing the representation of waves and the basis of the 
wave theory of matter. 
The historical background of wave-particle duality- 
the particle-like properties of light exhibited (for 
example) in the photoelectric effect, their parallel in 
the wave character of particles as suggested by deBroglie 
and demonstrated by diffraction of electrons, and finally 
the flowering of quantitative quantum m e c h a n i c ~ i s a 
familiar and rather romantic subject. The entire 
period of its invention and development was so short 
that many, perhaps most, of its major figures con- 
tributed to its very origins and were still aliveand active 
when it had become a mature and universally accepted 
cornerstone of physical science. Moreover the logic of 
its growth and the relationships between different 
theories and between experiment and theory is ex- 
ceedingly clear. And fortunately it is very well 
documented; consequently we shall make no attempt to 
discuss this background here. A few of the author's 
own favorite references are given in the bibliography. 
II. Matter Waves 
Let us examine some of the relationships associated 
with matter waves and with waves in general. For, 
after all, atomic orbitals are nothing more than the 
forms msumed by the standing waves characteristic of 
single electrons hound in an atomic potential field. 
We can discuss them i~?&igently and correctly only if 
we are ready to recognize and use their wave-like 
properties. 
To begin, let us distinguish standing waves from 
running waves. The latter are worth a little of our 
attention here for two reasons. First, running waves 
are less cumbersome in a development of the relation- 
ship between the quantum conditions and the wave 
equation. Second, the transition between running and 
standing waves gives us a natural and mathematically 
simple example of the principle of superposition, in its 
most precise way-in terms of the interaction of two 
readily visualized waves. 
8 It is worth noting that occasionally, respectable attempts are 
made to find classical models that fit better with quantum 
mechanics than the Bohr atom. One in fact appeared in pre- 
liminary form quite recently [GRYZINSKI, M., Phy8 Rev. Letters, 
14,1059 (1965)l. 
Traveling waves are appropriate for describing 
free particles; running waves, for describing bound 
particles. Any mathematical function of two or more 
variables-the time, t, and the spatial coordinate, x, 
for example--describes a running wave if the indepen- 
dent variable or variables can he written in one particu- 
lar form. If a function f(x,t) can be written as f(z) 
where z = kx - wt-that is, if x and t are always related 
so that the real independent variation off always can 
be given in terms of such a a-then the function f is a 
traveling wave. It need not be a periodic function like 
siu[A(kx - at)], hut it may be. The crucial property 
is this: a given point on a plot of f(z), say f(ao), cor- 
responds to an infinite number of pairs of values of x 
and t , corresponding to the solutions of kx - at = a. 
with k and w fixed by f. Since t moves inexorably for- 
ward, the value of x that keeps the quantity kx - ot, 
called thephase, equal to zo must also move inexorably 
forward, a t just the velocity w/lc, the phase velocity. 
Note that we have not introduced any explicit concept 
of wavelength or frequency because we have not yet 
discussed periodic waves. 
Periodic waves, and sinusoidal waves in particular, 
play a unique role in wave mechanics. Their mathe- 
matical simplicity, and the fact that combinations of 
sine and cosine waves can be used to represent any 
smooth function to an arbitrary degree of accuracy, 
make sinusoidal waves attractive. However, the 
property that makes them so important for atomic 
physics and chemistry is the fact that they give the 
exact representation of the simplest stationary states 
that occur in quantum mechanics, the states of a free 
particle. Let us see how these waves are a consequence 
of the Einstein relation between energy E and frequency 
v in cycles/sec [ (2~) - ' X o radians/sec], 
E = hv, (1) 
The deBroglie relation 
pA = h (2) 
between momentum p and wavelength A, and the 
classical expression for the total energy of a free par- 
ticle, is simply the kinetic energy: 
E = pn/2m (3) 
From relations (I), (Z), and (3) we obtain the con- 
nection between the frequency and wavelength of matter 
waves. tbthdispersion relation 
often written in terms of 1c = 2s/A instead of A, so that 
where 15 = h/2a. This is quite a diierent thing from 
the relation for light waves 
Free matter waves have a frequency that varies in- 
versely as the square of the wavelength, not as the first 
power--or alternatively, the phase velocity of matter 
waves in free space varies inversely with wavelength. 
If matter waves can be described by a wave equation, 
then we can infer that equation from their dispersion 
relation, eqns. (4a) and (4b). A wave equation descrih- 
ing a wave function fi is a relationship between the 
284 / Journd of Chemical Education 
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Highlight
time and space derivatives of J.. What is that relation- 
ship? 
In the function J.(x,t), the quantity u is a frequency, 
and must be associated with the time t to give a diien- 
sionless variable of the form vt. Similarly X must be 
associated with x to give a dimensionless variable 
x/X, or lcx. Since v appears to the first power, the wave 
equation must involve only the first derivative of J. 
with respect to time. The wavelength X appears as 
C2, SO that it must he the second derivative of J.(x,t) 
with respect to x which is related to the first time deri- 
vative: 
More specifically, using eqn. (4a), we have 
What about the numerical constant of proportionality? 
If it were m1, then J. would be a real exponential of the 
form e x p [ i (2svt + kx)]. This is a formal solution to 
the equation but is not quite acceptable in polite 
physical company because of its annoying property of 
becoming infinite when the exponent becomes large. 
A physically allowable free-state solution is one whose 
amplitude is at least bounded everywhere; this is 
readily achieved if our constant is i = 47. We have 
which has the solutions 
+ ( z , t ) = A e W - 2 4 (6) 
(We choose +i and not -i so that positive k corre- 
sponds to a wave moving to + m as t increases.) The 
form (6) for J. allows us to identify three differentiating 
operations with the evaluation of physical quantities: 
so that the time rate of change of J. for any x is a 
constant proportional to the energy of the state: 
next, the slope of the function J. is a constant propor- 
tional to the momentum: 
giving us the momentum p, and finally, we recognize 
that the identity 
is just equivalent to the dispersion relation, and 
amounts to writing 
E = p2/2m 
I n the general case, we identify ikb/bt with the total 
E-the sum of T, the kinetic, and V, the potential, 
contributions. 
Finally we conclude this little exposition of running 
waves by writing the running wave, eqn. (6), in its 
equivalent forms 
+(z , t ) = A [cos ( k z - 2rut ) + i sin ( k z - 2rv l ) l (9a) 
It is the last of these which is most important for our 
understanding of standing waves and atomic orbitals. 
Standing waves are waves that oscillate in time but 
whose crests and troughs remain fixed in space. For 
example $(x,t) = f(x) sin(2s vt) is such a function; at 
any point xo, $(x,t) oscillates between f(xa) and -f(xo). 
The function (9) is not a standing wave, but a super- 
position of them. We can rewrite (9) as the sum of 
+ ( z , t ) = A [cos Zsvt cos k z + i eos 2 n d sin kz + 
sin Zrvt sin k z - i sin 2sut cos k z ] 
four separate standing waves having just the right 
phases to give one real and one imaginary running 
wave moving together, as (9a) displays them. 
The mathematics that make a superposition of 
standing sine and cosine waves into a traveling wave 
are quite clearly exhibited in eqns. (9) and (10). At 
this point one could discuss the more philosophical 
aspects of superposition. One might, for example, ask 
about the probability of finding the electron in a sin kx 
distribution, if we know that the wave function is J.(x,t) 
of eqns. (9) and (10). We shall not pursue this point 
here [cf. Reference (5) 1. 
The point we must make now is this: the propor- 
tionality of v and E, and therefore the proportionality 
of the first time derivative and E, require that all 
stationary (constant E ) solutions of the quantum me- 
chanical wave equation for matter waves have a factor 
e ~ l r i u ~ . I n other words, the functions representing all 
the stationary states of an electron (or of a complex 
system) contain a complex oscillating factor containing 
the time. If the system is free, then the spatial part 
may be complex also, and the function J. can be a run- 
ning wave. 
The foregoing aside on running and standing waves 
has served two functions. It has developed in a sort of 
painless way a primitive example of the mathematical 
statement of the superposition principle. This princi- 
ple is the very basic quantum mechanical concept that 
there are always alternative and equivalent descriptions 
of an electron wave. No one description necessarily 
tells us explicitly all the properties of the electron that 
we might want to know, or makes apparent all the 
useful ways of interpreting the physical properties of a 
wave function. 
The concept of superposition will become a very 
important one in our discussion of alternative repre- 
sentations of orbitals; of hybridization; and of valence 
bond, molecular orbital, and mixed representations. 
The basic concept to be grasped now is the existence of 
equivalent descriptions, any one of which can be ob- 
tained from any other by a re-expression or transforma- 
tion no more subtle or complicated in principle than the 
transformation that gives us eqn. (9) as an alternative 
to eqn. (6). 
The second main reason we have dwelt on the con- 
cept of a wave is to develop the time dependence of an 
electron wave in its simplest example. Now, as we 
proceed into a discussion of bound states and of atomic 
orbitals, we will be using a form and a physical picture 
that will let us drop the explicit time dependence of our 
wave function. Nevertheless, all along, it is important 
to remember that every wave has a time dependence, 
that electron waves describing states of constant energy 
have factors and therefore have real and imaginary 
Volume 43, Number 6, June 1966 / 285 
Usuario
Highlight
parts whose amplitudes oscillate sinusoidally in time 
about mean values of zero. 
We conclude this general discussion by amplifying 
briefly the physical and mathematical notions associ- 
ated with the concept of a stationary state, and with the 
idea that a wave function should have a factor e""'. 
Let us generalize eqns. (5) and (7) a bit for our later use 
by supposing that E is not necessarily p2/2m hut may 
also contain some potential energy V(x) that may or 
may not vary with distance hut does not vary with 
time. Then, instead of eqn. ( 5 ) , the general statement 
of (7) is 
or, by way of definingthe Hamiltonian X, 
If the Hamiltonian X does not contain time explicitly, 
and ours clearly does not, then X acts only on the space 
variable of $(x,t) and not on t. But the partial time 
derivative acts only on the time part. These two 
conditions can be satisfied only if itia$/at and XJ . are 
one and the same constant multiple of J. itself. The 
multiplicative factor is obviously just E, from our 
previous discussion. But this implies that the energy 
E is constant in time, i.e., that the energy is stationary, 
or that the system is in a statiaar?~ state. "Stationary" 
in this usage does not imply that the wave function is 
constant, but only that the energy is constant and the 
wave function is a periodic traveling or standing 
wave. The second implication follows from the fore- 
going because the form of eqn. (11) implies that J.(x,t) 
can he written as a product +(x)t(t), and that 
and 
The function t(t) is simply e-""'; +(x) is just a func- 
tion of x independent of time, so that J.(x,t) is neces- 
sarily a wave with period Elti. 
Where does the idea of discrete quantized states enter 
our physics? So far, all we have done applies to con- 
tinuous distributions of states. The discrete quantiz* 
tion is, in essence, a result only of the introduction of 
finite boundary conditions. So long as we make no 
restriction on how the wave function behaves as it goes 
off to * m, there is no quantization. (We require 
only that the free functions remain bounded by some 
upper and lower limit.) However as soon as we intro- 
duce "finity" boundary conditions-like saying that 
+(x) must vanish a t the walls of a box located a t *a, or 
that +(x) must correspond to a function on a ring and 
+(x) = +(x+211), or that +(x) must go to zero expo- 
nentially as x approaches hoth * m -any such condi- 
tions immediately remove the possibility of a continuum 
of E values and of a corresponding continuum of states. 
[A detailed discussion of this is given in Ref. (,$).I In 
essence, the imposition of houndary conditions a t both 
ends of the range, plus the conditions that the hound- 
state wave have no kinks or discontinuities and he 
quadratically integrable, eliminates all possible func- 
tions except those having an integral number of half- 
waves within some region (not arbitrary) appropriate 
to each individual problem. All other formal solutions 
would in some way fail to satisfy the various conditions. 
Ill. Discrete Stationary States for Single 
Particles: One-Electron Systems 
Orbitols in One Electron Atoms 
Under what circumstances do we find discrete 
quantized states for a single particle? These circum- 
stances are the kind that lead to the quantized states of 
the particle in a square box, the harmonic oscillator, the 
rigid rotor, and the one-electron atom or molecule. The 
circumstances require that the potential energy V have 
a dip or well of some sort, so that V(m), its value at 
infinity, is higher than its value somewhere in the well. 
If there are any states of the particle whose energy E is 
less than V(m), then these states must be discretely 
quantized. For example, the simple harmonic oscillator 
with V = kx2/2 has infinite V(m), so that all its states 
are quantized. The hydrogen atom has V(m) = 0, 
conventionally, so that any state of negative energy, E 
< 0, must be discretely quantized. 
Graphically, the discrete quantization is a generaliza- 
tion of the discrete quantization of the oscillatory states 
of a rope with hoth ends fastened, or of a particle in a 
box. For the rope or particle-in-box, a continuum of 
oscillations is possible if only one end is held or if one end 
of the box is open. However, if both ends are fastened or 
closed, respectively, the only oscillations are those that 
give constant displacements of zero at hoth ends (see 
Fig. 1). If the rope's length is L, these have the spatial 
form A sin (nrrx/L) (or some combination of these) 
where n is any positive integer. The generalization 
nodal point 
(b ) 
Figure 1. I.) Rope with a free end; the dirplocement of the end of the 
rope con hove any volue. The oniy conditions are that ot one end (x = 01, 
the dirplasement y(O1 = 0, and the1 the other end geh no further from 
x = 0, y = 0 than 1, the lengthof the rope-i.e., ylendl 5 1. 
(bl Rope with both ends swashed, the conditions y101 = 0 and yIL1 = 0 
allow the rope oniy a discrete (but infinitel number of vibrational stoles, 
nomely thwe with 0, 1.2. ... nodes between the ends. 
comes when we replace the two rigid fastenings with ex- 
ponential decreases to zero a t both ends, + m , for all 
three Cartesian coordinates-or tie the two ends of the 
rope together. 
It is worth noting that a central potential may be 
attractive and still not be able to s u ~ n o r t bound states. 
In the region where E < V, the wave function is curving 
away from the axis, so that i t must be leaving the axis 
as it enters the potential well. In the region where E > 
V, and the wave function is curving back toward zero 
amplitude, its curvature is always proportional to the 
depth of the well below its energy value. If the well is 
both shallow and narrow, the wave may be unable to 
bend back to re-enter the forbidden region with its 
slope inclining toward the axis, the slope required to 
make the function die exponentially as it penetrates 
the region where E < V . If no wave can re-enter 
properly, then no wave can correspond to a bound state. 
Figure 2 illustrates this behavior; we can picture it in 
terms of trying to fasten a very stiff rope to two hooks 
in a tight space. 
Bound States in a Potential 
Figure 2. Three types of behavior; (a) bound, quantized rtote with curva- 
ture just suitable far matching both decaying curves with the &wroidd 
curve; ib) and (4 phygically impossible situations, corresponding to no true 
states; the wove function with E < V is only o dying exponential on one 
ride, and grows exponentidly on the other. Arrows mark points where 
E = Vand cvrvotvrechonger sign. 
The exponential decrease is a very simple consequence 
of the mathematics whenever the total energy E is less 
than the potential V. We refer again to Ref. ( 2 ) , 
pp. 51-58, for a particularly clear exposition of this 
topic. It is worth noting one point now that will become 
especially important in the last section of this paper, 
dealing with electron correlation. Whenever E < V, 
the kinetic energy is necessarily negative, and the momen- 
tum, being p2/2m, becomes imaginary. This seems 
strange and formal, but it is just as strange, formal, and 
above all nonclassical to have E < Vat all. The exist- 
ence of the wave function in nonclassical regions is 
one of the most important physical differences between 
classical and quantum mechanic^.^ 
With our conclusions from the previous section and 
the paragraphs just preceding, we have a basis for a 
clear but still qualitative picture of a stationary state of 
a singleelectron atom. The electron is described by a 
complex wave in three dimensions; the wave's real and 
imaginary parts oscillate sinusoidally in time, 90' out 
of phase, with a frequency v = E/h. The energy E 
of the electron is negative, and can only assume cer- 
tain discrete values, and the wave itself goes to zero 
exponentially as r, the electron's position vector, goes 
to infinity. Such a function is the simplest example of 
an atomic orbital. 
Quantum Numbers and Constants of the Motion 
So far we have characterized an atomic orbital by 
one number only, the energy. We all know perfectly 
well that there are other characteristic numbers or 
quantum numbers associated with orbitals. Let us 
inquire into their origin. 
That E is a good quantum number is a consequence 
of the conservation of energy. This is a rather trivial 
statement when it is put this way, but we can say it 
slightly differently: if X, the Hamiltonian of a system, 
does not change explicitly with time, then the energy of 
that system will be a constantor a good quantum num- 
ber. That means that if the potential and the pa- 
rameters such as mass, and the universal constants do 
not depend on time, so that X a t t , is identical with X 
a t ts then E will be a good quantum number. 
The other familiar quantum numbers of atomic 
orbitals represent other physical constants of the 
motion associated with other invariances of a Hamilton- 
ian, i.e., of the physical description of a system. The 
quantum number 1 of a particle is a good quantum 
number when the total orbital angular momentum of 
the particle is a constant. This comes about if, and only 
if, the Hamiltonian of the system is spherically sym- 
metric. Since the kinetic energy depends only on 
momentum and makes no reference to any spatial co- 
ordinate, i t is as symmetrical in space as it can be; 
we need only examine the potential energy for its 
symmetry. If it depends only on r, the distance of the 
particle from the origin, and not on any angle, then 
naturally the potential energy is spherically symmetric. 
Suppose we are studying an electron bound by a 
potential V. The electron has some instantaneous 
angular momentum. We now seize the apparatus that 
produces V, and rotate the apparatus to a new angular 
orientation without translating it a t all. If the potential 
were not spherically symmetric, such a motion would 
clearly disturb the electron, in general, but if V were 
spherically symmetric, rotating the apparatus would 
leave the electron entirely unaffected. Specifically, if 
V is not spherically symmetric, the rotation would in 
general introduce a torque on the electron and change 
its orbital angular momentum. If V is spherically 
symmetric, then the electron's orbital angular momen- 
tum is unchanged by the operation applied to V. 
In other words the orbital angular momentum is a con- 
stant of the motion. 
For the sphere, or for a spherically symmetric system, 
Q mmacroseopie example of the penetration of a. quantum 
mechanical wave through a classiedly forbidden barrier zone i3 
the Josephson effert. In this phenomenon current emtrrien in 
semi- and super-conductors are able to penetrate layers of insnla- 
tor between two semiconditcting or mpereondneting bodies. 
Volume 43, Number 6, June 1966 / 287 
re-orientation into any angle is a symmetry operation: 
re-orientation leaves the system in a condition in- 
distinguishable from its initial condition. Symmetry 
operations never come singly (except those for the most 
unsymmetrical things we can think of, things that can 
only be left alone) and do not come in arbitrary com- 
binations; in general one operation followed by another 
is equivalent to a third operation. The set of all opera- 
tions associated with a particular symmetry type is the 
grmp of operations for that symmetry. The set of all 
rotations constitutes the simplest full symmetry group 
for a spherical system. The rotations about the figure 
axis constitute a symmetry group for a cylinder or a 
helix. 
Just as the total orbital angular momentum is a 
constant if V is spherically symmetrical, the angular 
momentum component about the figure axis of a cylin- 
drical system is a good quantum number. The in- 
variance of the cylindrical system with respect to any 
rotation about its axis implies the existence of a constant 
of its motion, the corresponding component of orbital 
angular momentum. 
A special case of a system with cylindrical symmetry 
is the sphere--and, indeed, a spherical system has a 
constant orbital angular momentum component along 
an arbitrarily chosen axis, as well as a constant total 
orbital angular momentum. This quantum number is 
m,, of course. (We shall sometimes use m for m,.) 
Why can we not h d the angular momentum com- 
ponents along three axes of a sphere, instead of just one? 
At the risk of being too brief, we can answer this just 
by saying that such knowledge would localize the 
particle's orientation too much, to the point of violating 
the uncertainty principle. 
In general, in a spherically symmetric potential, the 
energy of a hound electron depends on its angular 
momentum-i.e., on 1, as well as on the principal 
quantum number n, but never on m. (The exception is 
the Coulomb potential, for which all states of the same 
n are of equal energy, regardless of I.) There are 21 + 1 
values form, for any 1, so that there are 21 + 1 different 
degenerate orbitals having the same n but different m 
"-1 
[n2 or C (21 + I), for the Coulomb or hydrogen-like 
1-0 
case]. These 21 + 1 different states transform into 
mixtures of each other if we redefine the orientation of 
the coordinate axes of our spherical system. But no 
matter how we re-orient the coordinates, a given set of 
21 + 1 wave functions transform only into each other, 
and never into any other functions. This is exactly 
analogous to the way sin no and cos n8 can be trans- 
formed by addition and subtraction into en' and e-"O 
or mixtures of these, but never into anything containing 
e-"n+l)e. The set of 21 + 1 functions is called a basis 
for a representation of the rotation group. 
I n fields of lower symmetry than spherical, the 
orbital angular momentum is no longer quantized. 
However, the symmetry invariances may remain in 
part. Sometimes we can suppose that 1 is a good, con- 
stant quantum number but m is not (Russell-Saunders 
coupling: there, the same thing happens to spin; S is 
a constant but M, is spoiled as a quantum number). 
In other cases, even 1 is not constant. It is very often 
useful to start with a set of free atom functions, either 
orbitals or many-electron state functions, with all the 
degeneracy appropriate to a spherical potential; then 
one asks what happens to this particular set of orbitals 
if the symmetry of the potential is lowered to something 
less than spherical, like octahedral or simple threefold 
for example. One uses the symmetry properties of the 
wave functions to determine their behavior in a t best a 
semiquantitative way. This can be an exceedingly 
powerful way of elucidating chemical problems. The 
most famous application of this sort is of course crystal 
and ligand field theory in its phenomenological form. 
We shall return to this topic for some examples, but we 
shall not try to develop the entire theory of group 
representations in atomic physics and chemistry. 
Several references a t various levels are given in the 
bibliography. The pertinent conclusion for us now is 
this: if the symmetry of the potential V is lowered from 
spherical to some new form, then some but not all the 
degeneracies of the spherical case may remain. The 
rather straightforward algebra of group representations 
lets us determine very easily exactly how any given basis 
set of 21 + 1 degenerate functions from the spherical 
case will split into smaller sets in the new and lower 
symmetry. The answers are expressed in terms depend- 
ing only on 1 and on a smaller number (sometimes only 
one) of angle-independent quantities that can be treated 
as empirical parameters or can be evaluated from atomic 
wave functions. We shall return briefly to this subject 
later. 
The Forms of One-Electron W a v e Functions 
The mathematics of a spherically symmetric one- 
electron problem lead directly to a set of standing waves 
that we can understand and describe very easily, a t 
least in part. Putting together the pieces of the forego- 
ing discussion, we can tell just what to expect. In gen- 
eral, the characteristic stationary waves must describe 
constant-energy states whose total orbital angular 
momentum is characterized by 1, and these waves must 
come in degenerate sets of 21 + 1. If we make use of 
all the spatial constants of motion, then these functions 
correspond to the states of definite mi. We are free to 
describe a system of energy E in terms of any combina- 
tion of m, eigenfunctions. As we have indicated 
previously, it is sometimes useful to suppose that the 
appropriate wave function is notan eigenfunction of 
any one component of angular momentum, but has 
some other property, like directionality. 
The solution of the spherically symmetric Schro- 
dinger equation appears almost as soon as we convert 
the Hamiltonian into a form that can be written as a 
sum. In the sum, one set of terms contaiosall the radial 
dependence including the entire potential, and the other, 
all the angular part, which is only kinetic. (We let 
3, = rZT, and 3 0 , ~ = r2 X l(1 + 1)RZ/2mr2, or rZ X 
Tn,* rotational energy.) 
r'X = [3, + +V(T)] f %.p (14) 
r-dependent angledependent 
only only 
We say X is separable when it can he so expressed, as 
a sum of independent terms. Using the same reasoning 
that gave us equations (12) and (13) from (l l) , we 
obtain a product form for fi(r, 8, q) : 
288 / Journal of Chemicol Education 
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and one equation for each of the three factors R(r), O(8) 
and %(q). The function R(r) must depend on the 
specific problem. It is known, naturally, for the 
Coulomb potential V(r) = -e2/r, and for the general 
Ar-" potential as well. I n the next section we shall 
see how such a problem arises and is treated in many- 
electron atoms. The Coulomb solutions are given by 
Laguerre functions; some of the lower ones are shown 
in elementary texts. 
Basically, all the radial solutions R&) for bound 
states have certain properties in common. They all go 
exponentially to zero as r goes to infinity; the function of 
lowest energy is nodeless and each successive higher 
function with the same 1 has one more node than the one 
before it, and the functions are orthogonal and can 
always be normalized in the sense that 
6 R,:*(r) R,,i(r) r2d~ = 
S,.,' (i.e., 0 if n f n', and 1 if n = n') 
These conditions have the usual consequence that each 
function reaches its maximum amplitude in its outer- 
most lobe, and that each successively higher function 
has its outer lobe a t larger r than the one before. This 
is all clear and obvious in the case of the hydrogenic 
functions, but is is worth noting that these character- 
istics are quite general. I n the next section we show 
some radial functions for atoms. 
The angular functions are the very well known spher- 
ical harmonics, the forms taken by standing waves in 
any spherical problem. For example the standing 
waves on a flooded planet exhibit exactly the same 
angular behavior as do atomic orbitals hut occur only 
on a single spherical surface, so they are perhaps easier 
to visualize than the wave function of an atomic elec- 
tron. This example is developed in Ref. (3). 
Table 2. Some Lower Spherical Harmonics Y d 8 . d 
s-function: 1 = 0 
pfunctions: 1 = 1 
Y P = & eos 8 
d-functions: 1 = 2 
~ ~ - 1 = 415 sin 0 cos 0 ec's = 
8s 
4z sin 20 e'v 
- 
yZs = $3 ,in2 6 = !+5 (1 - eels 28) e2tq 
3% 2 32% 
Table 2 represents some of the analytical expressions 
for the spherical harmonics Ytm(8, q) = O(O)%(q). 
Certain general properties are worth explicit mention. 
First, we see that each Yim(8,u) is a polynon~ial of de- 
gree 1 in sin 8 and cos 8. These can be expressed, alter- 
nately, as trigonometric functions of multiples of 8. 
Either form lets us visualize the 8-dependence explicitly. 
We could combine functions like YI1 and Y1-I to get 
real (i.e., not complex) comhinationsvarying as sin 8 cos 
a (i.e., as x) and as sin 0 sin q (i.e., as y). 
Another point to recognize about the spherical 
harmonics is a symmetry property with respect to the 
origin. The functions with even 1 all contain only 
even powers of sin 8 and cos 8, and the odd 1 functions, 
only odd power. If we change s to -x, y to - y, and 
z to - z (i.e., invert the coordinate system through the 
origin), then Y's of even 1 go into themselves (even or 
gerade functions) while Y's of odd 1 go into their nega- 
tives (odd or ungerade functions). 
Third, as the number of angular nodes increases with 
I, so each lobe becomes more directional or pointed. 
Very high 1 functions, with large numbers of angular 
nodes, describe electrons that are nearly classical in 
their orbital rotations. For, according to the Corre- 
spondence Principle, when the wave length of an elec- 
tron wave is small compared with the dimensions of the 
volume in which the electron moves, the particle 
ceases to show its wave character and behaves like a par- 
ticle. 
By contrast, the radial parts of the one-electron bound 
state wave functions retain quantum character even for 
high quantum numbers because the outermost lobes are 
always both the largest and the most spread out. The 
inner parts of highly excited states, the parts at low r 
values, do have rapid oscillations. Consequently, in 
states of high n, electrons behave classically in their 
radial coordinate when they come near the nucleus but 
as waves when they move to large r and their local 
momentum is small. 
IV. Atomic Orbitals in Many-Electron Atoms 
The Many-Elecfron Problem 
Up to this stage, we have examined the wave func- 
tions for a single electron moving in a potential field, 
especially a spherical potential. But most atoms do not 
consist of a single electron moving in a potential. Let us 
now explore the way one can develop well-defined wave 
functions for single electrons in many-electron systems. 
Then we can look at some of the properties of these 
orhitals and see how they are used in some representa- 
tive chemical problems. In the next and final section 
we can look at the limitations of the orbital method, 
and see how its limitations affect our interpretations of 
physical problems and how we can try to overcome these 
limitations. 
Suppose we consider first the two-electron case, the 
helium atom. The lowest state of He must have both 
electrons in 1s orbitals, we say. But the Hamiltonian of 
the system contains potential terms -ez/r,, -eZ/rz, and 
e2/rlz--that is, not only is the nuclear attraction for each 
electron part of the potential; the electron-electron 
repulsion is part also. Obviously a t any instant neither 
electron moves in a spherically-symmetric potential. 
Then how can we possibly refer to a 1s orbital, much 
less assign both electrons to it? 
The rationale for assigning the electrons in a qualita- 
tive building-up model of the atom and the method by 
Volume 43, Numher 6, June 1966 / 289 
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which we calculate the shapes of orbitals depend on the 
notion that we can find some effective potential V(r,) 
for each electron. Each V(rj) must have spherical 
symmetry and approximate the true potential felt by 
electron j . Could such an effective potential be found, 
and could one solve the resulting equations to find 
orbitals for a many-electron atom? 
Both questions have been answered with an almost 
positive "yes." The proper effective potential was 
developed by Hartree, Fock, and Slater in the early 
1930's for atoms with closed shells or with one electron 
or one hole in a closed shell. Their original method 
left some ambiguities about the best way of defining 
V(rJ for arbitrary open-shell atoms. Then, the method 
was exteuded by several people so that potentials could 
be defined for open shells. (These can probably never 
be made to approximate real potentials as well as the 
Hartree-Fock potentials for closed-shell atoms. This is 
inherent in the slipperiness of an open shell system-the 
electrons tend to move simultaneously from one m, 
state to another within a given open shell of lixed 1. 
The mathematics of open shells tells us this when it 
requires that a single stationary state be a superposi- 
tion of several specific assignments of electrons to mi- 
states, i.e., to be a multideterminant function, if we 
may use a term to be defined later.) 
The Hartree-Fock method consists of using a 
Hamiltonian for each electron that contains V(r,) de- 
fined by the Hartree-Fock prescription, which we shall 
describe shortly, and solving all the equations togetherfor all the electrons of an atom. The original solutions 
for HartreeFocB orbitals were obtained by numerical 
integration and were therefore tabulated functions with 
no analytical expressions. Numerical solutions are 
still obtained and, at their most refined level, are prob- 
ably still the most accurate. Analytical approxima- 
tions for Hartree-Fock orbitals can be obtained with 
high-speed computers; they are extremely useful and 
often are very close approximations to the best func- 
tions we have. 
Note that when the method reaches the stage of com- 
putation, the Hartree-Fock equations are ordinary, not 
partial differential equations. This is because the 
Hartree-Fock potential is chosen to be spherical, so that 
the orbitals must he spherical harmonics multiplied by 
radial functions. These radial functions are the solu- 
tions of the differential equations. The symmetry of 
the problem allows it to be reduced this way by letting 
us use our general knowledge about spherical systems 
to go most of the way. 
The seemingly mysterious potentials of the Hartree- 
Foclr equations are defined this way: the potential for 
each electron is the mean potential due to the nucleus 
and all the other electrons-more specifically, it was 
shown to be the root-mean-square potential. The orig- 
inal form used by Hartree utilized a simple Coulomb 
field based on Born's probabilistic interpretation of the 
wave function. If $(r) is the amplitude of the wave 
function at r, then '$(r) or $*(r) $(r) is the inten- 
sity of the wave there, a quantity everywhere posi- 
tive and whose space integral is normalized to unity. 
These properties led to the identification of $(r)i2 as the 
density of probability, or, in our minds replacing a time 
average with a space average, led to the identification 
of l$(r)i2 as the mean charge density at r. Hence each 
electron must feel the Coulomb field of the spherical 
average or the spherically symmetric sum of the charge 
densities defined by all the i$(r),2's for all the other 
electrons. 
But, we ask, how can we find one orbitd unless we 
know all the others? The answer to this question is the 
crux of the success of the method. We can start with a 
guess for all the Vsof the system, with a most outlandish 
collection of orbitals if we wish. We use all but one of 
these to determine the potential for the last electron, 
and then solve the differential equation for this last 
electron's orbital, $(N). Then we use the new orbital 
with all the originals but one, say $(N-,), to determine a 
refined version of the deleted $(N-I) from the original 
set. The two new orbitals $(N) and $(N-l) plus the old 
ones give us a third new one, $(N. 2), and we continue 
until we've determined an orbital for each electron. 
Then we start through the process again, refining our 
first $(,,, $(,-I), . . . , $cl) revisionsinto second revisions, 
and then go through again and again until we find that 
the orbitals are not changed by further recycling. The 
potential field defined by this series of operations is now 
consistent with the orbitals that it determines and that 
determine it. It is called a self-consistent field or SCF, 
in this case a Hartree SCF. 
Exclusion Principle and Many-Electron Wave Functions 
We must interject a brief acknowledgment of the 
existence of electron spin and a review of its role in 
many-electron problems. (It could have been part of 
our discussion of symmetry; cf. References (36) and 
(27).) And we must introduce the Pauli Exclusion 
Principle, too. The former adds the quantum number 
112, to our set n, 1 and m, for a oneelectron function; 
strictly, it adds s as well, but since all the electrons we 
know haves = '/s, we don't bother carrying it explicitly. 
We do have to pay attention to m, (= the com- 
ponent of s along one chosen axis, in any many-elec- 
tron or magnetic field problem. Them, quantum num- 
ber's values do contribute very much to magnetic prop- 
erties; more germane for this context, spin adds one 
extra degree of freedom to each orbital. 
A spatial function $(r) with an assigned m, of +'/2 
is said to be a spin orbital with or spin, and if it is neces- 
sary to designate the spin state explicitly, is usually 
written either as $(r) a or as $(r) alone. If m, is 
assigned as one commonly writes $(r)B or &(r). 
Sometimes one need only say that $ should stand for 
the entire spin orbital. 
The Pauli Exclusion Principle is the statement that 
no two electrons may be assigned to the same spin 
orbital. This can be restated many other ways, per- 
haps less concisely: no more than two electrons can be in 
the same orbital, and if two are in the same one-electron 
state in coordinate space, they must have different spin 
states; or, no two electrons in a system can have the 
same values for all their quantum numbers. If two 
electrons with the same spin are forced together in 
space, they must go into different energy states. One 
may remain in a low-energy state, but the other must go 
to at least the next higher energy orbital. Such a re- 
striction is equivalent to the requirement that the two 
electrons stay apart in momentum space if their wave 
functions are close together in coordinate space. The 
measure of this requirement is closely related to the 
290 / Journal of Chemical Education 
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uncertainty relation ApAq 2 R/2 for any single coor- 
dinate of one particle. The requirement, stated in 
terms of the six-dimensional space of three spatial 
coordinates and three momentum coordinates is simply 
that each electron needs a volume AT = AxAyAz 
Ap,Ap,Ap, = h3. 
Electrons are identical particles, so that no physical 
property can be affected if we rename or renumber them. 
For example (and not an accidental example), if we deal 
with a wave function q(rl, . . . , rN) for N electrons, then 
-?(r,, . . . , rN)= must be unaffected if we interchange two 
electrons; moreover the wave function *(rl, . . . , rN) 
must go back into itself if we interchange the same pair 
of electrons twice. These two properties imply that 
interchanging two electrons must either leave Y un- 
changed ( q is symmetric) or, at most, change it into its 
negative ( q is antisymmetric). Now \E must always he 
zero if two electrons are assigned to the same spin 
orbital, according to the Pauli Principle. This is just 
what the situation would be if Y were to change into 
- q whenever a pair of electrons was interchanged. If 
electrons 1 and 2 were arbitrarily assigned to the same 
spin orbital, then the identity of electrons and their 
spin orbitals would imply that -?(r,, r,, . . , r = 
( r r , . . ., rw) but also our guessed permutation 
property would require that q(r,, r2, . . ., rN) = 
-V(r%, rl, . . ., rN), so that this Y would necessarily be 
zero. We have never seen a nonzero Y in nature with 
two electrons in the same spin orbital; we can infer, 
therefore, that the antisymmetric choice correctly 
describes the behavior of real many-electron functions. 
This argument is not meant as a rigorous proof of the 
antisymmetric property of many-electron functions but 
only as a demonstration of the relationship between the 
Exclusion Principle and the property of antisymmetry. 
The Form of Many-Electron Orbital Wave Functions 
Now we can go on to consider the form of many- 
electron wave functions and the relationship of this form 
to atomic orbitals, and then we shall return to the 
Hartree-Fock problem. 
If we assume that we can find appropriate self-con- 
sistent potential fields V(rJ for each electron of an 
atom, then the Hamiltonian of the atom can be written 
as a sum of Hamiltonians, each one containing the 
kinetic energy T, and potential energy V(r,) for only 
one electron. That is, 
x = C xi 
sll electrons 
j 
Once again, as withequations (12) and (13) and with (14) 
and (15), we have a separable Hamiltonian. This time 
it is separable into its one-electron terms. As before,whenever the Hamiltonian is separable, the wave 
function can be expressed as a product of functions, 
each of which is the solution of its own equation: 
XAr;) = v&;) (17) 
That is, we may write 
*(r,,. . . r ~ ) = +(11(11). . . I L I N ~ I N ) (18) 
where we let each indicate the J t h spin orbital. 
Equation (18) is a very useful form and tells us very 
explicitly something about the kind of atomic wave func- 
tion we get if we start with the orbital concept. This 
equation says that in this picture, the electrons have 
probability amplitudes, and therefore probability dis- 
tributions, that are independent of the position of any 
other electron. The only effect elertron 1 has on 
electron 2 is through mutual effect of their average 
potentials of interaction with each other and to a lesser 
degree, with the other electrons. 
Expression (18) cannot present the oomplete picture 
even in terms of orbitals because it does not have the 
property of being antisymmetric with respect to ex- 
change of any pair of electrons. If we exchange elec- 
trons 1 and 2, and require that the function q be re- 
paired so that it changes sign with this exchange, then 
we can do it this way: replace (18) with 
Now we can make any other pair interchange and sub- 
tract the corresponding new functions from each of the 
two terms in (18a) to get to a more repaired state. 
Eventually we'll reach some of the rearrangements 
again; in fact we can construct all the permutations of 
the N electrons among the N spin orbitals &,). If we 
continue to change the sign when we make a pair ex- 
change, we will eventually construct the totally anti- 
symmetric function that can be constructed with the 
spin orbitals $ill, . . . , fiIwl. The factor 1 / 4 2 in (18a) 
was added to keep the function normalized. When we 
have all N permutations, the normalizing factor is 
I/.\/#!!, instead of 1 / 4 2 . 
I t was pointed out by J. C. Slater that the construc- 
tion of a completely antisymmetric function from a set 
of products, adding and subtracting as we have just 
described, is exactly equivalent to making a determinant 
out of the J.ln(rk) spin orbitals. One lets the number 
(J) of the spin orbital be the column index and the elec- 
tron number k be the row index, or vice versa. 
In this way we have 
This expression is the Slater determinant, or the deter- 
minantal function based on the spin orbitals $(I,, . . . , 
\I.(N). Because Slater determinants are so commonly 
used, and because it is really redundant to write more 
than the principal diagonal or the N spin orbitals them- 
selves, once we know that a determinant is meant, we 
frequently use a shorthand: 
Even the normalizing factor is left implicit. Another 
common notation applicable even if \Ir is not a product 
function uses the antisymmetrizing operator (2 (which 
is usually defined to do the normalizing also). We say 
simply that if (2 acts on the function (IS), it generates 
the function (19). Although this seems like an ar- 
bitrary and rather useless formal definition, it is possible 
to write out explicit prescriptions for (2, and to use this 
Volume 43, Number 6, June 1966 / 291 
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shorthand as a powerful tool for dealing with many- 
electron problems. 
It turns out that a single determinant function like 
(19) is a very suitable form for a Hartree-Fock wave 
function whenever all the spin orbitals of a given n and 1 
are either full or empty-i.e., in the closed shell cases 
like He, Be, Ne, or Zn+2. The form is also fine when 
there is one electron outside a closed shell structure or 
one hole in a closed shell system. In other cases, if we 
try to use a single Slater determinant to define the wave 
function of N electrons, we find we are in trouble. 
Except for special cases, we cannot obtain functions 
with quantized total orbital angular momentum L or 
total spin S with single determinant functions. Physi- 
cally, in the proper functions with quantized L and 8, 
the inl and in, values of the individual electrons in the 
open shells are no longer good quantum numbers. The 
correct function based on orbitals must be written as a 
sum of two or more determinants. Here are two 
examples. The lowest triplet state of He has orbitals 
1s and 2s each singly occupied. In the notation of 
(19a), we have 
The extra 1 / 4 2 is another normalizing factor. The 
two assignments in (20) differ only in assignments of 
m,. In effect, the electrons flip each other's spins. A 
more drastic reorganization takes place in the 8P ground 
term of the carbon atom: 
Similarly the 2p2, 'D, and '8 levels of carbon also in- 
volve 1 and s reorganization (we omit 1s and 2s elec- 
trons in writing). 
P(p:'D,Mfi = 0, MI = 0) = 
1 - 1212~0%l + 12p+%I - l%+2~- I I, and 
*(p,= '8, Mr. = 0, M. = 0) = 
1 -- 43 [ PP+% I - 1 2 x 2 ~ - l - 12~02x1 I 
In this example, the 2p electrons change both their m. 
values and their m:s. In all cases of this sort, the 
changes are not random but occur in concert. Here, 
by the way, we have our first explicit inkling that there 
might be a limitation to the orbital picture, just be- 
cause in the second of these examples, we cannot assign 
a specific set of four quantum numbers to the 
electrons. We can specify only n and I for each elec- 
tron. 
The set of states defined by assigning n and I for each 
orbital is called a configuration. A real state is based on 
a single configuration just so far as each one-electron 
nand 1 are true constants of the system. 
If, in addition to all then's and l's, we specify L and S, 
then we are said to specify a term, like ls22s22p2 to give 
aP of carbon. From any single configuration we can 
always find one or more terms. 
What physical effects result because we must use 
the antisymmetrized function (19) instead of the simple 
product (18)? We have discussed how the anti- 
symmetry property was associated with the Exclusion 
Principle and with the fact that the electrons can't all 
condense in the 1s orbital of an atom. 
The antisymmetry property exhibits its effects on the 
forms of atomic orbitals in a very explicit way. This 
is through the famous exchange interaction which Fock 
added to Hartree's simple average Coulomb interaction 
in the self-consistent field. In Hartree's original ex- 
pressions, the potential energy of electron j in an atom 
had the form 
Fock showed that an antisymmetric function like 
(19) introduced a similar-looking but physically different 
term, due to single pair permutations: 
Because of the additional pair exchange, these al- 
ways appear in closed shell cases as the combination 
FJK - GJK. Also the G's are zero if J.c,l and $yK1 have 
different m,, while the F's are not. The effect of GJK is 
to reduce the effect of Coulomb repulsion between elec- 
trons of the same spin in $qJ, and J . ( K ) . This in turn 
permits the orbitals to be closer to the atomic nucleus 
and therefore to be more strongly bound. The im- 
provements in energies with the exchange interaction 
included are quite dramatic; the effects on wave func- 
tions are sometimes subtle. Figure 3 shows a compari- 
son of typical Hartree and Hartree-Fock radial func- 
tions. 
Hartree-Fock orbitals have been computed numeri- 
cally to high accuracy for some atoms, and approximately 
for the entire periodic table. They have also been 
computed in analytic form to rather high accuracy for a 
number of atoms. The very first analytic form for the 
radial wave function, historically, was the simple ex- 
ponential, introduced in 1930 by Frenkel. The most 
popular extension of this form is a monomial multiply- 
ing an exponential: 
The parameter 3 is fixed by normalization, so that 
only a is a free parameter. These are the Slater type 
orbitals; Slater's original set of orbitals was based on a 
simple set of rules for choosing optimum ru's [J. C. 
SLATER, Phys. Rev. 36,507 (1930)l. The most popular 
of the modern accurate analytical formsis a sum of 
functions like (24), often with several terms of the same 
n. The other popular form in current use is based on a 
sum of Gaussians, which replace ar or (24) with j3rBr2. 
These have the advantage that they lead to simpler 
integrals than do exponentials in r, but more of them 
are required to reach a given level of accuracy. Refer- 
ences to some of the more extensive tables and sources 
are given in the bibliography (4f-44). 
The Hartree-Fock orbitals, as we said, represent the 
best ae+lectron functions based on the mean field of all 
the electrons, in the sense of giving the best one-elec- 
tron energies in this mean field. Because of this energy 
292 / Journol of Chemical Education 
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property, Hartree-Fock orbital energies give very good 
estimates of ionization potentials (and electron affini- 
ties, if we start with Z + 1 electrons around a nucleus of 
charge +Ze). The orbital energy of the highest filled 
orbital is approximately equal to the ionization energy. 
I t was also shown that Hartree-Fock orbitals give good 
estimates of all other atomic properties that involve in- 
teractions of a potential field without inducing transi- 
tions. By "good," we mean accurate to about the same 
percentage accuracy as the total energy. Typical 
properties that are evaluated well by Hartree-Fock fnnc- 
tions are charge distributions, electric field gradients a t 
nuclei (the source of nuclear quadrupole coupling in 
chlorine compounds), and in certain cases like the alkali 
atom principal series, spectral line frequencies and in- 
tensities. Other properties that are likely to be less 
accurately described by Hartree-Fock functions are 
properties like spin-spin and spin-orbit coupling, that 
depend on electron-electron interactions, and spectral 
frequencies and transition probabilities that involved 
more than a simple s-to-p transition, like the forbidden 
atmospheric line of oxygen. 
One property for which there is a significant number 
of calculated and experimental values is the ordinary 
electric dipole polarizability, the induced dipole per 
unit of applied electric field. Table 3 contains some of 
the calculated (Hartree-Fock) and experimental polar- 
izahilities for several atoms and ions. It appears that 
the lower the polarizability, the lower the relative error 
of the calculation, which is probably because the species 
with high polarizabilities have many low-lying excited 
states including doubly-excited states or states that are 
not included in the Hartree-Fock calculation. 
Table 3. Experimental and Theoretical (Hartree-Fock) 
Dipole Polarizabilities 
Atom or Experjmentd~ 
Ion Theoretical (A3) (Aa) 
COHEN, H. D., J . Chem. Phys. 43, 3558 (1965). 
L:ANGHOFF, P. W., AND HURST, R. P., P h ~ s . Rev. 139, A1415 
(1965). 
Experimental values compiled by A. Dn~GanNo, Advan. Phys. 
11, 281 (1962). 
Transformations of Orbitals: Hybrids and Ligand Field 
Orbitals 
One of the most striking properties of molecules is 
the relative rigidity of their structures. If we had no 
information about their structures, we might well 
attribute much looser structures to them, structures 
more like liquid drops. It has been one aim of theore- 
tical chemistry to understand the directional character 
of chemical bonds, and to do this in terms of atomic 
orbitals, if possible. 
One of the most attractive ways to interpret the 
directional character of bonds is based on using direc- 
tional atomic orbitals. These are the hybrid orbitals 
in a free atom. I n a molecule, one can use the same 
hybrids, or the closely related equivalent orbitals, or 
any of several other similar localized orbitals; they vary 
a little in their definitions and manner of construction 
but prove to be quite similar in the long run. We shall 
here restrict ourselves to atoms, sidestepping a lengthy 
discussion of these various localized molecular orbitals. 
Let us first examine the mathematics relating the 
Hartree-Fock orbitals to the localized orbitals. This 
way we can see how the best approximations to one- 
electron states of constant energy are related to the best 
approximations to localized orbitals for electron pairs. 
Then we shall compare the two types of orbitals. 
The approach is based on a transformation like one to 
which we alluded in our discussion of spherical har- 
monics. The localized orbitals appear when we super- 
pose the standing waves that are Hartree-Fock orbitals. 
We quote and then use a theorem about our deter- 
minantal functions like (19): we can make any unitary 
transformation we like of the filled orbitals and leave 
the value of the determinantal function (19) unaffected. 
Recall that a unitary transformation is like a pure ro- 
tation of a coordinate system: it preserves all the lengths 
(normalization) and angles (orthogonality) when it 
transforms the old coordinates (functions) into the new 
ones. This means that if we mix the individual factors 
+(rJ with each other by adding and subtracting them 
in any size pieces, so long as we maintain the ortho- 
gonality and normalization of the new $'s, the deter- 
minant (19) formed from the new +'s will be equal to 
the old determinant, for any fixed choice of electron 
coordinates. We can mix functions with 1 = 1 and 
mr's of +1 and - 1 to get functions that look like sin 0 
cos p and sin 0 sin p in their angular dependence; that 
is we can go from 
R(n)R(n) sin Ole+ sin B d w - 
R(r,)R(rl) sin A e - i ~ , sin B& 
to 
R(?,)R(m) sin 8, eos o, sin 82 sin m - 
R(r,)R(m) sin 8, sin q: sin & cos oz 
by letting 
= .@ [Rfr) sin B sin ql 
4a 
We have transformed from complex orbitals with 
quantized mr into directional orbitals looking like cos 0 
but with respect to the z and y axes. 
The orbitals can be made still more directional if we 
are willing to spoil not only mr but also 1 itself as a good 
one-electron quantum number. We can mix a spheri- 
cally-symmetric s-function with a cosine-like p-function; 
if we mix them by adding equal parts of each, then on 
the positive z-axis, where cos 0 has its maximum, the p- 
wave will add its maximum to the outer (positive) lobe 
of the s-function and reinforce the total wave ampli- 
tude. On the negative z-axis, the p-wave will tend to 
Volume 43, Number 6, June 1966 / 293 
cancel the s-function, or, in optical terms, there will be 
maximum destructive interference of the waves. If 
the radial parts of the s and p-waves are very similar, as 
they are if n = 2 for both waves, then the reinforcement 
and cancellation is very significant. In fact, for this 
case, the p-wave slightly more than cancels the s-wave 
on the negative z-axis. Written out, we have (using the 
normalizing factors from Table 2), 
With equal parts of the normalized s and p-functions, 
we have constructed an sp hybrid. If we had sub- 
tracted the p-function from the s-function, we would 
have simply changed z for -z and put the reinforce- 
ment onto the negative z lobe, and the cancellation on 
the positive side. 
In this example, we constructed two equivalent sp 
hybrids. We could do the same to construct three 
equivalent sp2 hybrids from two p's and an s. Let us 
do this example with p, and p,-orbitals, keeping to the 
x, y plane, and use the fact that p-orbitals are likevec- 
tors in their angular behavior. First, one may be 
chosen as 2/3 pU: 
Now, the other two functions must contain A; they 
must contain equal parts of J.nz with opposite signs and 
these parts must contain all of J.,, (i.e., the sums of the 
squares of the coefficients of each component orbital 
must add to 1); and they must contain all the rest 
of J.vn divided evenly and with the same sign, so 
that both of them point toward the negative y direction. 
This fixes the other two as 
and 
For convenience in illustration, one sometimes im- 
plicitly supposes that Rz0(r), the 2s radial function, is 
identical with the 2p radial function Ral(r), so that the 
angulardependence of the hybrid orbital can be plotted 
unambiguously in one or more planes. In truth, Rzl(r) 
and Rza(r) are not generally identical, as Figure 3 shows. 
What relationship exists between the localized hy- 
brid orbitals or localized orbitals in general, and the 
Hartree-Fock orbitals that we examined previously? 
First of all, the two sets give the same total energy of 
any atom, by virtue of our quoted theorem. Second, 
they can be obtained from similar but by no means 
identical sets of equations. The Hartree-Fock orbitals 
are in one sense the best orbitals we know how to find- 
the energetic sense. The Hartree-Fock equations and all 
the fundamental equations we have for deriving orbitals 
ab initio (not just by transforming them into each 
other) are variational equations. They say that the 
one-electron energy of each orbital is the lowest possible 
energy that we can find for it, so long as each orbital is 
orthogonal to the other orbitals of the same atom that 
have still lower energies. The only condition the 
Hartree-Fock orbitals satisfy, other than the minimum 
energy and orthogonality conditions, is the normaliza- 
tion condition, to conserve our scale of probability. It 
was shown only recently that localized orbitals, even 
the most localized orbitals in the sense of maximum 
intra-orbital interaction of electron pairs, can also be 
solutions of well-defined and rigorous variational equa- 
tions. This is an important point historically and 
didactically. It seemed for quite some time that since 
there were physically-based and mathematically sound 
equations for Hartree-Fock orbitals, they rested on a 
firm basis while the localized orbitals were in a more 
vulnerable situation, without any rigorous physical 
basis. They were always widely used, hut with some 
trepidation about their physical interpretation. In 
any event, the variational equations for localized 
orbitals necessarily carry other conditions than the 
energy condition of the Hartree-Fock equations. There 
is avariety of conditions one can use, like the one requir- 
ing that the electron-electron mean interaction within 
an orbital be maximized (and therefore the inter- 
orbital mean interaction becomes minimized). 
r(Bohr radii) 
Figure 3. Radio1 functions RIr) for neutral carbon in in ground state. 
The solid curves ore Hortree-Fosk functionr UUCYS. A., Proc. Roy. Soc. 
(London1 A173, 5 9 (1939)l. m d the dotted curve is a Hortree 2p function 
[TORRANCE, C. C., Phyr. Rev. 46, 3 8 4 1193411. The Hortree Is and 2s 
functions are very rimiior to the Hortree-Fock functions .how" here. Note 
the rimilority of the outer parts of the 2s and 2 p functions. (The functions 
plotted are mluolly R = r +, ro that J P R'dr = 1 .I 
The equations for localized orbitals are somewhat less 
tractable than those for Hartree-Fock orbitals. If we 
are only interested in visualizing and describing localized 
orbitals qualitatively, it is relatively easy and satis- 
factory to use existing Hartree-Fock orbitals to make 
hybrids within a specific shell. Symmetry alone fixes all 
the mixing coefficients for us and we can derive them 
with a little algebra, just as we did in the previous sec- 
tion. However if we want the localized orbitals that are 
best by some criterion like the one we just cited, then 
we are forced to do a moderate amount of computation. 
294 / Journal of Chemicol Education 
The variational equations for localized orbitals are more 
cumbersome than the Hartree-Fock equations, so that 
i t may be best to find the Hartree-Fock orbitals first and 
scramble them to meet the extra conditions. 
Naturally, since localized orbitals must meet extra 
conditions, they cannot be the best orbitals in the 
orbital energy sense. Nor can they satisfy the sym- 
metry conditions, the conditions that lead to the spher- 
ical harmonic form for each Hartree-Fock orbital. Nor 
can they have quantized angular momentum. (It may 
be that for open shell atoms, the energetically-best 
Hartree-Foclc orbitals don't have quantized angular 
momenta either. This question is currently unsolved.) 
Moreover the fact that localized orbitals do not give the 
best one-electron energies means that they do not give 
good values for ionization potentials, a t least from one- 
electron energies. Of course if we compare the total 
energy of atom and corresponding positive ion to get the 
ionization potential, it makes absolutely no difference 
whether we use the Hartree-Fock functions or some 
local set obtained by scrambling the Hartree-Fock 
functions. However, the Hartree-Fock orbital energies 
are generally better approximations to ionization 
potentials than the total energy differences. 
It may turn out that localized orbitals are more u s e 
ful than Hartree-Fock orbitals in other ways. One 
hope is that they will offer a straightforward way to 
evaluate correlation energies. Correlation energy in an 
atom or molecule is the difference between the actual 
total (usually with relativist,ic contributions subtracted) 
and the Hartree-Fock total energy. It represents the 
extra energy of the system coming from the fact that 
electrons really interact with real electrons, not with 
mean fields of other electrons. It would be helpful 
if it turns out that we can calculate correlation effects 
by looking a t electrons just two at a time. If that is the 
case, then localized orbitals are likely to be extremely 
useful for this purpose. As yet, we cannot say for sure. 
V. Beyond Orbitals-The Correlation Problem 
As we just said, the Hartree-Fock energy differs from 
the true energy of an atom or molecule by a relativistic 
contribution that need not concern us here: and a 
correlation contribution. The origin of the correlation 
energy is quite clear and we can even attribute to it a 
precise potential field, the Jluctuation potential. This 
total field is the difference between the mean field 
and the exact field felt by an electron. An example 
is shown in Figure 4. In a two-electron atom one can 
visualize this field clearly. It acts between two elec- 
trons but depends somewhat on the distances between 
electrons and nuclei. It is highly repulsive and very 
short-ranged, stronger when the two electrons are far 
from the nucleus than when they are close. When one 
electron comes sufficiently close to the other, the 
electrons' potential energy becomes greater than their 
potential and, correspondingly, their kinetic energy 
'This is not to say that relativistic effects are negligible. In 
terms of the total energy, relativistic effects become important 
even in the second-row elements. However, these contributions 
occur primarily in the inner shells, part,icols;rlp because of the 
high kinetic energies associated with the inner shells. As a result, 
the relativistic effects have little direct effect on the chemical 
behavior of atoms. 
must become negative and their momenta, imaginary. 
This is a two-particle, three-dimensional example of 
the phenomenon we discussed in connection with the 
existence of quantized bound states of a single particle. 
Figure 4. The fluctuation potentiol for I s electrons in beryllium (after ref. 
26). The fluctuation potentiol is the difference between the exact and 
mean potentidr. It is o function of n 0% well as of rs; the value r~ = 0.27 
a,"., the most proboble Is rodiur, w m chosen. The figure described po- 
tentiolr dong the linecontaining the nucleus and electron l . 
5 
We should interject a comment on the quantitative 
importance of the correlation problem. These energies 
in atoms are just the size of chemically important energy 
differences-roughly 1 or 2 ev per pair of opposite-spin 
electrons in the valence shell and higher for shells nearer 
the nucleus. Moreover correlation is the primary 
source of London forces, and therefore of the cohesive 
energy of molecular crystals. I n other words, even 
though we may get qualitative, graphic, and clear no- 
tions about electronicwave functions from orbital 
descriptions, and even though we may find that orbital 
calculations are useful for correlating chemical phe- 
nomena, we must be exceedingly cautious about using 
any wave function for quantitatiu~ purposes unless it con- 
tains correlation effects or unless we can show that the 
correlation effects drop out in our particular problem. 
The tendency of one electron to repel another and to 
force the wave functions of the two electrons to have low 
amplitudes when the electrons are near gives rise to the 
concept of the correlation hole. This hole takes the 
form of a region around each electron, the region where 
the fluctuation potential is very large, where no other 
electron is likely to be. The Pauli Exclusion Principle 
establishes this hole moderately well for electrons of the 
same spin, but it has no effect on the spatial distribu- 
tion of electrons with opposite spins, so does not help 
to introduce any correlation effect. I n this case the 
correlation hole must be a pure Coulomb hole and can 
only be introduced into a wave function by inclusion of 
specific terms above and beyond the Hartree-Fock func- 
tion. I n other words we can improve on the Hartree- 
Fock function by superposing additional terms, which 
are the result of the fluctuation potential, to account for 
electron correlation. 
What does the fluctuation potential do to our orbital 
concept? One thing it cannot do. I t cannot spoil 
Volume 43, Number 6, June 1966 / 295 
4- 
ex~cttw-elsctmn Wtentiol 
e2/r,, when electron 1 is of 
- 
3- 
averope potentiol seen by 
2- 
-3 -2 -I t 
7 7 I 
6 - 
- 
I 
I 
I 
I 
I I 
Bcryilium 1s electrons - - fluct~ation potential 
- *en by electron 2 
the overall symmetry or angular momentum of our total 
wave function. It can, however, spoil virtually all the 
other bases of our orbital picture. First of all, it ob- 
viously spoils any quantization of the angular momen- 
tum and energy of individual electrons. This means, 
in turn, that individual electrons cannot he described 
by specified quantum numbers n and 1. This is turn 
means that we cannot, strictly, specify the configura- 
tion of an atom or molecule. The entire structure 
of our atomic physics seems momentarily to be crashing 
down. 
I n fact, the situation is not disastrous. To a reason- 
able degree of approximation, we can specify atomic 
configurations and the n and 1 quantum numbers 
of individual electrons. (We generally cannot specify 
individual m(s and m,'s though; the electrons' inter- 
actions do spoil their orientations.) At least we can 
specify the most likely or most important atomic con- 
figuration, or one-electron n,l and orbital energy. 
This brings us directly to the problem of how one 
can actually go beyond the orbital picture, and how one 
can get a better wave function than the Hartree-Fock 
function. There are two principal approaches. The 
first historically and the most extensively explored is 
the method known as configuration interaction. The 
second has several variations, but all are based essen- 
tially on the concept of cluster expansions. 
Configuration interaction is, as its name suggests, the 
addition to the Hartree-Fock N-electron function of one 
or more similar functions having different assignments 
of the individual n's and 1's of the electrons. These new 
functions correspond then to configurations differing 
from the Hartree-Foclc configuration. For example the 
beryllium Hartree-Fock configuration is obviously 
1~2292. We could imagine adding to this some lsZ2s2p 
and ls22s3s. However neither of these will do: the 
first is necessarily an overall P state (total L = 1) while 
our Hartree-Fock and true functions are both S states, 
and in the long run, we can only build the latter with S 
configurations. The second, ls22s3s, does not affect 
construction of the lowest state provided we have really 
used Hartree-Fock functions. Strictly, the ls22s3s can 
enter to a very slight degree, but for a first and very 
good approximation, we can neglect it. I n fact 
we are justified in saying that neither it, nor any other 
configuration that differs from the one of interest in the 
quantum numbers of a single electron, will contribute 
to configuration mixing with the configuration of in- 
terest. Inotherwords the ls22s2configuration beryllium 
mixes with ls22p2 and ls23s2 but not with 1sZ2s2p in the 
overall zero-angular momentum IS state of the atom. 
What physical effect is associated with this configura- 
tion interaction? The basic idea is relatively easy to 
see. Let us neglect the two 1s electrons. The 2s radial 
function of Be looks roughly like that of Figure 3 and 
the angular function is of course constant. In the 2sZ 
configuration, the radial and angular coordinates of one 
electron are entirely unrelated to the radial and angular 
coordinates of the other. Now, if we add some 2p2 
character to the total wave function, we introduce some 
correlation between the two sets of angular coordinates. 
The 2p2 part has the effect of accounting partially for 
the polarization of one electron by the field of the other. 
This occurs because the two electrons' p functions rein- 
force their s functions in different angular regions. 
When one electron's probability amplitude is large in a 
particular direction as a result of constructive inter- 
ference of its s and p parts, the other electron's proh- 
ability amplitude is largest about 90' away, due to its 
own simultaneous interferences. The admixture of 3sZ 
has a similar effect on the radial variables. In effect, 
mixing 3sZ with 2s2 introduces some in-out correlation 
(cf. References 52-55). 
Configuration interaction is a formally exact method 
that must converge to the exact wave function of the 
Hamiltonian of choice. I t is computationally straight- 
forward and we are learning to interpret the physical 
significance of its various sorts of terms. In general, it 
is a very slowly converging proce~s.~ The beryllium 
atom was treated with 37 conf?gurations, for one of the 
most accurate calculations available on any atom more 
complex than helium. Configuration interaction is 
probably the method of choice at present if one simply 
wants a very accurate wave function of a moderately 
complex atom and cares very little about computational 
efficiency. 
The other general approach to correlation has its 
origins in the fact that one can treat the interaction of a 
single pair of electrons. One can get a much more 
accurate representation of helium-like a t o m than one 
gets from the orbital model alone. This better rellre- 
sentation comes from the addition of terms in the two- 
electron wave function that contain the interelectronic 
distance explicitly. One of the simplest examples is a 
dying exponential factor e- ""'?' that can be included 
as a factor only for a small range of the inter-electronic 
distance r12 around the value rlz = 0. A much more 
elaborate one is the wave function of He originally 
introduced by Hylleraas, which contains a number of 
terms in r12 is its most highly developed form. 
The general approach to correlation through pair 
interactions can be paraphrased somewhat generally 
this way. I n atoms, it may be valid to suppose that the 
order in which one should calculate interactions is: 
(1) interaction of each electron with the mean field of all the 
others (HartreeFock); 
(2) deviations from (1) due to simple pair interactions 
(binary encounters in two electrons in the HartreeFock field of 
the rest); 
(3) deviations from (1) and (2) due to three-body encoun- 
ters; etc. 
If binary encounters are much more important than 
three-body effects, then we should be able to treat part 
(2) by using methods similar to those developed for the 
two-electron problem. This approach n-as developed 
by Szasz, Tsang, and Sinanoglu, and is perhaps most 
graphically described in an article by Sinanoglu (51). 
The results indicate that for atoms up to beryllium,